Lemma 15.81.7. Let R \to A \to B be finite type ring maps. Let M be a B-module. If M is finitely presented relative to A and A is of finite presentation over R, then M is finitely presented relative to R.
Proof. Choose a surjection A[x_1, \ldots , x_ n] \to B. Choose a presentation
A[x_1, \ldots , x_ n]^{\oplus s} \to A[x_1, \ldots , x_ n]^{\oplus t} \to M \to 0
given by a matrix (h_{ij}). Choose a presentation
A = R[y_1, \ldots , y_ m]/(g_1, \ldots , g_ u).
Choose h'_{ij} \in R[y_1, \ldots , y_ m, x_1, \ldots , x_ n] mapping to h_{ij}. Then we obtain the presentation
R[y_1, \ldots , y_ m, x_1, \ldots , x_ n]^{\oplus s + tu} \to R[y_1, \ldots , y_ m, x_1, \ldots , x_ n]^{\oplus t} \to M \to 0
where the t \times (s + tu)-matrix is given by a first t \times s block consisting of h'_{ij} followed by u blocks of size t \times t given by g_ iI_ t, i = 1, \ldots , u. \square
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